I would go with the 2nd option **(checkboxes)** because:

 - **Radio buttons** usually express **mutually exclusive options** (*EITHER this OR that*), so giving an option that includes an item but excludes it at the same time **won't be consistent**.   
Contrary to checkboxes it's  expected to *select exactly 1 item, neither less nor more*.
 
 - **Checkboxes** are used **to select several items** (*this AND that*) which is the case. Also they let you select **none**. Bear in my that it might happen that you have to add a *new option* and if you use radio buttons, then "both" will be meaningless and you'll have to redesign this control (also remember *users hate changes unless they're great improvements*).
 - **"Both" option has more cognitive load since it doesn't express something by itself**, it's a reference to another items, so the *state is not directly showed*. Additionally you are adding +1 element to the UI. A way to improve it would be to explicitly name the label "Customer + Item" or "All", similarly as the case of the classic multiple choice questions, where you can find options like "*Answer D*: All previous answers are correct" or "A and C are correct".   
But anyway, if the answers are not mutually exclusive, you could add a "All" checkbox as a shortcut for users, without losing *visibility of the selection state*. There's a similar approach used in table-like components, like [Gmail Inbox][1], where you have the "select all" checkbox. And [in this question][2] you have an example using "None" with checkboxes too.


  [1]: http://images.pcworld.com/images/article/2012/07/gmail20is20unimportant-11383765.jpg
  [2]: http://I%20would%20go%20with%20the%202nd%20option%20**(checkboxes)**%20because:%20%20%20-%20**Radio%20buttons**%20usually%20express%20**mutually%20exclusive%20options**%20(*EITHER%20this%20OR%20that*),%20so%20giving%20an%20option%20that%20includes%20but%20excludes%20at%20the%20same%20time%20won't%20be%20consistent.%20%20%20%20Contrary%20to%20checkboxes%20it's%20%20expected%20to%20*select%20exactly%201%20item,%20neither%20less%20nor%20more*.%20%20%20%20%20%20%20%20-%20**Checkboxes**%20are%20used%20**to%20select%20several%20items**%20(*this%20AND%20that*)%20which%20is%20the%20case.%20Also%20they%20let%20you%20select%20**none**.%20Also,%20it%20might%20happen%20that%20you%20have%20to%20add%20a%20*new%20option*%20and%20if%20you%20use%20radio%20buttons,%20then%20%22both%22%20be%20meaningless%20and%20you%20will%20have%20to%20redesign%20this%20control%20(also%20remember%20*users%20hate%20changes%20unless%20they're%20great%20improvements*).%20%20%20%20%20%20%20%20-%20**%22Both%22%20option%20has%20more%20cognitive%20load%20since%20it%20doesn't%20express%20something%20by%20itself**,%20it's%20a%20reference%20to%20another%20items,%20so%20the%20*state%20is%20not%20directly%20showed*.%20Additionally%20you%20are%20adding%20+1%20element%20to%20the%20UI.%20A%20way%20to%20improve%20it%20would%20be%20to%20explicitly%20name%20the%20label%20%22Customer%20+%20Item%22%20or%20%22All%22,%20similarly%20as%20the%20case%20of%20the%20classic%20multiple%20choice%20questions,%20where%20you%20can%20find%20options%20like%20%22*Answer%20D*:%20All%20previous%20answers%20are%20correct%22%20or%20%22A%20and%20C%20are%20correct%22.%20%20%20%20But%20anyway,%20if%20the%20answers%20are%20not%20mutually%20exclusive,%20you%20could%20add%20a%20%22All%22%20checkbox%20as%20a%20shortcut%20for%20users,%20without%20losing%20*visibility%20of%20the%20selection%20state*.%20There's%20a%20similar%20approach%20used%20in%20table-like%20components,%20like%20[Gmail%20Inbox][1],%20where%20you%20have%20the%20%22select%20all%22%20checkbox.%20%20%20%20%20%20%20%20[1]:%20http://images.pcworld.com/images/article/2012/07/gmail20is20unimportant-11383765.jpg%20%20%20[2]:%20http://ux.stackexchange.com/questions/67609/adding-a-none-of-the-above-option-to-a-list-of-checkboxes/67616#67616